## Damage Modeling

Pneumatic valves can exhibit many different fault modes. External leaks of the pneumatic gas can develop at the pneumatic ports, and internal leaks can arise between the two valve chambers. Further, friction can build up over time as the lubrication wears down and particulate matter is formed due to the sliding motion of the piston. The spring can also degrade over time.

To capture these mechanisms in the model, we define the rates of change of system parameters using wear and damage accumulation equations. For example, friction in the valve is linear with the area of the asperities at the contacting surfaces. As sliding wear occurs, this area increases. Sliding wear volume increases linearly with the product of sliding force and sliding velocity. Therefore, we may express the friction parameter, r(t), in a similar form:

where wr is the wear coefficient, Ff(t) is the sliding force (in this case the friction force), and v(t) is the sliding velocity (in this case the valve velocity). The figure below shows the effect of an increase in friction on the valve cycle. From the simulation, we can determine at what value of the friction parameter that the valve has reached end of useful life (EOL), thus requiring maintenance. At this value, the friction force becomes large enough that the valve cannot open within the 15 s limit, as shown below.

Valve Operation with Increasing Friction

As the valve continues to cycle, the friction parameter takes on an exponential-like growth, as shown below. This is clear from the fact that the friction force is linear with the friction parameter, so as the friction parameter grows the friction force will grow, and this will increase the rate at which the friction parameter changes.

Evolution of Friction Parameter as a Function of Total Valve Cycles

Damage in the spring can be modeled in a similar manner, except using the spring force, so that the more the spring is used, the weaker it becomes:

where wk is the wear parameter, Fs(t) is the spring force, and v(t) is the valve velocity.

The figure below shows the effect of a decrease in the spring parameter on the valve cycle. In normal operation, without the spring tending the valve to close, the valve will open faster and close slower. However, the spring must be strong enough to close the valve against system pressure when the actuating pressure is lost. Through the simulation we can determine the maximum value of the spring parameter in which this functionality is retained. This defines the EOL with respect to this parameter.

An internal leak in the valve can also form as a result of sliding wear:

The mass flow of the leak is computed as

where the leak area is defined by Ai(t). The presence of an internal leak makes it more difficult to actuate the valve, because it causes gas to flow into the lower pressure volume that is being evacuated out of the higher pressure volume that is being filled. This is demonstrated in the figure below. With a large enough leak size, the valve cannot open within the 15 s limit.

Valve Operation with Increasing Internal Leak

External leaks can also form, most likely at the actuator connections to the pneumatic gas supply, due to corrosion and other environmental factors. We assume the growth of the area of the leak holes is linear:

The external leak rates are defined by

where patm(t) is atmospheric pressure.

The effect of the formation of a leak at the pneumatic gas connection at the top of the valve is shown in the figure below. As the leak grows, it becomes easier to open the valve but more difficult to close it. Through simulation we can determine the minimum size leak hole at which the valve cannot close within the 15 s limit, and use this to define EOL. EOL can also be defined based on prescribed maximum allowable leakage rates from the valve.

Valve Operation with Increasing External Leak at the Top Port

In the figure below we show the effect of a leak at the pneumatic gas connection at the bottom of the valve. The valve becomes more difficult to open but easier to close. Through simulation, we can determine the minimum size leak hole at which the valve cannot open within the 15 s limit.

Valve Operation with Increasing External Leak at the Bottom Port